Conformal mapping Definition and 49 Threads

Conformal Cyclic Cosmology, or CCC, is a hypothesis put forward by Roger Penrose in the early 2000s. My understanding of physics is lacking so my explanation will not be that clear, but I will summarize it here. Essentially, the existence of a previous spacetime, or "aeon," is postulated. This...

The question here is not asking for links to help understand analytic continuation or the Riemann hypothesis, but rather help in understand the bits of hand-waving in the following video’s explanations : https://www.youtube.com/watch?v=sD0NjbwqlYw (apparently narrated by the same person who does...

Hi PF! Does anyone know the conformal map that takes a wedge of some interior angle ##\alpha## into a half plane? I'm not talking about the potential flow, just the mapping for the shape. Thanks!

Hi, I'm studying about Conformal Mapping in Complex Analysis and see its applications in Heat transfer, Fluid and Static Eletrocity. But it is said that this subject is very useful in many branches of Physics. Can you tell me about that? Thanks.

Hello everyone: I studied in differential geometry recently and have seen a statement with its proof: Suppose there is a Riemannian metric: ##dl^2=Edx^2+Fdxdy+Gdy^2,## with ##E, F, G## are real-valued analytic functions of the real variables ##x,y.## Then there exist new local coordinates...

Homework Statement Find the images of the following region in the z-plane onto the w-plane under the linear fractional transformations The first quadrant ##x > 0, y > 0## where ##T(z) = \frac { z -i } { z + i }## Homework EquationsThe Attempt at a Solution [/B] So for this, I looked at the...

I know the concepts of conformal mapping and complex mapping but I didn’t see none explanation about how apply this ideia and formula for convert a curve, or a function, between different maps. Look this illustration… In the Cartesian map, I basically drew a liner function f(x) = ax+b...

I found this formula for doing a quadratic conformal map with parameters: I think there's probably a nice Einstein notation representation of this above but I haven't figured it out yet.. But anyway the mapping is like below: I don't know enough about General Relativity to know how this...

I am looking for conformal transformations to map: 1. Disk of radius R to equilateral triangular region with side A. 2. Disk of radius R to rectangular region with length L and width W. 3. Disk of radius R to elliptic disk with semi-major axis a and semi-minor axis b. Thanks!

"Definition: A map ƒ: A ⊂ ℂ→ ℂ is called conformal at z0, if there exists an angle θ ∈[0,2Pi) and an r > 0 such that for any curve γ(t) that is differentiable at t=0, for which γ(t)∈ A and γ(0)= z0, and that satisfies γ ' ≠0, the curve σ(t) = ƒ(γ(t)) is differentiable at t=0 and, setting u =...

Hi, I need to sketche ach of the following regions: R = {z :|z| < √2, 7π/16 < Argz<9π/16}, R1 = {z :|z| < 16, Rez>0} and write down a one-one conformal mapping f1 from R onto R1. Here is my sketch https://onedrive.live.com/redir?resid=4cdf33ffa97631ef%2110238 But I'm finding hard to find the...

What's a good textbook to learn conformal mapping in electrodynamics?

I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping. Anyone has any tips?

Is there a conformal mapping that transforms regular polygons (e.g. triangle and square) to circle?

Homework Statement With The map f(z) = -(1-z)/(1+z) where z=x+iy, and f(z) maps z onto w = u + iv plane. show for which points of the z plane this map is conformal. Homework Equations The Attempt at a Solution I have read a lot about this subject, and I think I...

Homework Statement Find function that maps area between ##|z|=2## and ##|z+1|=1## on area between two parallel lines. Homework Equations The Attempt at a Solution I don't know how to check if my solution works for this problem? I used Möbious transformation...

Homework Statement Find a conformal mapping of the strip ##D=\{z:|\Re(z)|<\frac{\pi}{2}\}## onto itself that transforms the real interval ##(-\frac{\pi}{2},\frac{\pi}{2})## to the full imaginary axis.The Attempt at a Solution I tried to map the strip to a unit circle and then map it back to the...

Hi, Homework Statement I'd like to show that the mapping w=u+iv=1/z tranforms the line x=b in the z plane into a circle with radius 1/2b and center at u=1/2bHomework Equations The Attempt at a Solution z*w=1=(b+iy)(u+iv) → 1=|(bu-yv)+i(bv+yu)| → u2+v2=1/(b2+y2) Now, a circle with radius 1/2b...

I'm studying Landau's Electordynamics of continuous media and, although I like how succinct it is, sometimes it is too succinct! I'm having trouble with a particular passage, so I'll just try to summarize the section up until the part I don't understand. The topic at hand is electrostatic field...

Hi, Given that the flow normal to a thin disk or radius r is given by \phi = -\frac{2rU}{\pi}\sqrt{1-\frac{x^2+y^2}{r^2}} where U is the speed of the flow normal to the disk, find the flow normal to an ellipse of major axis a and minor axis b. I can only find the answer in the...

Homework Statement We have the conformal map w = f(z) = z + K/z. Prove this mapping is indeed conformal. Homework Equations z = x + iy A map w = f(z) is conformal if it is analytic and df/dz is nonzero. f(z) = u(x,y) + iv(x,y) The Attempt at a Solution df/dz = 1 - Kz^-2 =/= 0 for finite...

Hello folks, I am trying to find a conformal mapping transform function that maps the following region in z-plane into interior of a unit circle in w-plane: |z-i|<\sqrt{2}\text{ ...AND... }|z+i|<\sqrt{2} Many thanks in advance for help & clues. Max.

A theorm I took down in class says: Consider the analytic function f(z). The mapping w=f(z) is conformal at the point z0 if and only if df/dz at z0 is non-zero. However, if df/dz does not exist at that point z0, is that point still a conformal mapping? That would make the function...

hi, I need a conformal mapping that changes the superellipse to an easier shape. if anyone send me any helpful thing (relative article, idea) I will be so pleased.

Homework Statement part ii of http://gyazo.com/0754ea00b2a4ea4a4d171906f6bf28bf Answers http://gyazo.com/821f370c502cd20210925f8498d18fa1 Homework Equations I did part i. I had to spot that 1/(x+iy)^2 = 1/(x^2+y^2)^2... (I subbed y = y-1) is this a standard result? Should...

Describe the image of the strip $\{z: -1 < \text{Im} \ z < 1\}$ under the map $z\mapsto\dfrac{z}{z + i}$ So I know that $-\infty < x < \infty$ and $-1 < y < 1$. Then $$ \frac{x + yi}{x + i(y + 1)} $$ Now if I take the the line y = -1, I have $$ \frac{x-i}{x} $$ Then find out what happens...

Homework Statement The transformation z=1/2(w + 1/w) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface. (b) Use...

Homework Statement The transformation z=\frac{1}{2}(w + \frac{1}{w}) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface...

Hi all, Suppose there is a bump at the origin, is there a conformal mapping between the bumped half-space (y>|b-x|, |x|<b && y>0, |x|>b) and the flat upper half space (y>0)? Anyone has a hint? Thanks in advance. Regards, Tony

What I'm trying to do is to apply conformal mapping and map the area bounded by the x-axis and a line at 60 degrees to the x-axis to the region above the x-axis. I think the basic goal of what I'm trying to do is to map \pi/3 to \pi. My problem is I really have no idea where to go from there...

Hi all How to calculate the capacitance of a sphere-plane system by using conformal mapping? Thanks

Hello! Please I need some help with this: Is it possible to transform a circle into a rectangle? If so what would be the expressions of x' and y' in terms of x and y. Thank you in advance!

Can you tell me is my solution true of the next problem. Find center w_0 and radius R of the circle k, in which the transformation w=\frac{z+2}{z-2} converts the line l:\text{Im} z+\text{Re} z=0. Solution: 2 \to\infty -2i=(2)^*\to w_0...

Homework Statement map the function \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation} on some domain which contains z=e^{i\theta}. \theta between 0 and \pi Hint: Map the semicircular arc bounding the top of the disc by putting $z=e^{i\theta}$ in the above formula. The...

how do we describe the biholomorphic self maps of the multiply puncture plane onto itself? I mean C\{pi,p2,p3..pn} Plane with n points taken away. I wanted to generailze the result for the conformal self maps of the punctured plane, but I do feel these are quite different animals. I...

Homework Statement Hi all. We are asked to transform the shaded area in below figure to between two concentric circles, an annulus. Where these circles' center will be is not important, just transform the area to between any two concentric circles. As you see in figure, shaded area is whole...

Homework Statement I'm trying to find a function that map the exterior of a circle |z|>1 into the interior of a regular hexagon. Homework Equations The Attempt at a Solution I have tried mapping the exterior to the interior circle. Then mapping interior circle to the upper plane which then I...

Hi, my question is what mapping to use for the problem in the picture attached. I need to be able to find the potential distribution etc by mapping from the x-y plane (as pictured) to a straight lines plane capacitor, which would be pretty straightforward, but I can't find this map in any...

Homework Statement Hi all. I have seen a conformal mapping of z = x+iy in MAPLE, and it consists of horizontal and vertical lines in the Argand diagram (i.e. the (x,y)-plane). On the Web I have read that a conformal map is a mapping, which preserves angles. My question is how this...

Let L:=\{z:|z-1|<1\} \cap \{z:|z-i|<1\}. Find a Mobius transformation that maps L onto the sector \{z: 0< arg(z) < \alpha \}. What is the angle \alpha? no idea of how about to set up the problem The intersection of the two circles forms a lens shaped region L with boundary curves, let's...

Hi everyone, (I hope I'm posting it in the right place, please feel free to move this thread to the appropriate place) My high school graduation project is about the application of the theory of complex variables in physics. Specifically, I am learning about the complex potential, its...

I'd like to map the open unit circle to the open ellipse x/A^2 + y/B^2 = 1. How would I go about doing this? I really have no idea how to go about doing these mappings. I'm working with the text Complex Var. and Applications by Ward and Churchill which has a table of mappings in the back...

I heard that one can solve 2D problem with conformal mapping of complex numbers. Is it possible to use this method for 3D axial-rotationally symmetric problems (which are effectively 2D with a new term in the differential equation)?

Homework Statement "Study the infinitesimal behavior of f at the point c. (In other words, use the conformal mapping theorem to describe what is happening to the tangent vector of a smooth curve passing through c.)" f(z) = 1/(z-1), c=i Homework Equations |f'(c)| and arg f'(c)...

When trying to solve one problem (my own, not an exercise), I encountered the need for a conformal mapping between a square [0,1]^2 and a triangle (0,0)-(1,1)-(2,0), so that the side (0,0)-(0,1) of the square gets mapped into a point (0,0), and the three other sides become the sides of the...

Does anyone know what the conformal mapping of an aerofoil at incidence is? Does it use the Joukowski transformation? Or something else.. Thanks

Is it possible to transform an ellipse x^2/a^2 + y^2/b^2 = 1 ("a" minor or major semiaxis) Into a rectangle? If so, how can I do it? I am not very familiar so please explain all the details. I know the transformation from a circle to an airfoil, but not this one.

Hi everyone, Let me set the scene. I'm writing a program to model the flow of an ideal fluid around various singularities using the complex potential and then using conformal transformations to map boundaries into new shapes. It's very nearly done but one of the transformations (what appears...

I would appreciate if someone could explain Conformal Mapping using Complex Analysis using an example. I get the rough idea but have no clue how complex analysis comes into the picture. Thank You!